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Use one of the rectangles in the task, and ask students to find its perimeter and area. Ensure that students have a chance to process what these terms mean before they proceed with the rest of the activity.
If students struggle to come up with side lengths of rectangles that meet the constraints, encourage them to sketch quick diagrams of rectangles. Consider providing graph paper for this purpose.
If calculators are needed to help with numerical computations to access the task, provide them. If not, this activity is a good opportunity to practice some mental math.
Use Collect and Display to create a shared reference that captures students’ developing mathematical language. Collect the language that students use to describe the rectangles. Display words and phrases, such as “rectangle,” “perimeter,” “area,” “length,” and “width.”Here are some rectangles.
| length (units) | width (units) | perimeter (units) | area (square units) |
|---|---|---|---|
| length (units) | width (units) | perimeter (units) | area (square units) |
|---|---|---|---|
The goal of this discussion is to generalize how the dimensions of a rectangle affect its perimeter and area.
Direct students’ attention to the reference created using Collect and Display. Ask students to share their answers about Rectangles D and E. Invite students to borrow language from the display as needed. As they respond, update the reference to include additional phrases.
Focus discussion on any consistencies that students noticed when perimeter was held constant and they considered side lengths that resulted in different areas, and vice versa. Students may recall from earlier courses how the shape of the rectangle generally affects the perimeter and area. If they do not bring up this concept, draw their attention to it. Here are some questions for discussion:
Here are two tables. The first shows some measurements of Rectangle A, with a side length of 5 cm. The second shows some measurements of Rectangle B, which is a square.
Complete the table for Rectangle A, and be prepared to explain your reasoning.
| length (cm) | width (cm) | perimeter (cm) | area (sq cm) |
|---|---|---|---|
| 5 | 1 | ||
| 5 | 2 | ||
| 5 | 4 | ||
| 5 | 20 | ||
| 5 | 40 | ||
| 5 | 28 | ||
| 5 | 50 | ||
| 5 |
Complete the table for Rectangle B, and be prepared to explain your reasoning.
| length (cm) | width (cm) | perimeter (cm) | area (sq cm) |
|---|---|---|---|
| 1 | 1 | ||
| 2 | 2 | ||
| 3 | 3 | ||
| 4 | 16 | ||
| 8 | |||
| 100 | |||
Sketch the graph of each pair of quantities, where the width is plotted along the -axis.
and the perimeter of Rectangle A
and the area of Rectangle A
and the perimeter of Rectangle B
and the area of Rectangle B
The purpose is to describe the relationships explored in the tables and graphed. Here are some questions for discussion: